Lee classes on LCK manifolds with potential
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916559970107392 |
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| author | Ornea, Liviu Verbitsky, Misha |
| author_facet | Ornea, Liviu Verbitsky, Misha |
| contents | An LCK (locally conformally Kahler) manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $ω$ and a closed 1-form $θ$, called the Lee form, such that $dω=θ\wedgeω$. An LCK manifold with potential is an LCK manifold with a positive Kahler potential on its cover, such that the deck group multiplies the Kahler potential by a constant. A Lee class of an LCK manifold is the cohomology class of the Lee form. We determine the set of Lee classes on LCK manifolds admitting an LCK structure with potential, showing that it is an open half-space in $H^1(M,{\mathbb R})$. For Vaisman manifolds, this theorem was proven in 1994 by Tsukada; we give a new self-contained proof of his result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03363 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lee classes on LCK manifolds with potential Ornea, Liviu Verbitsky, Misha Differential Geometry Algebraic Geometry 53C55, 32G05 An LCK (locally conformally Kahler) manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $ω$ and a closed 1-form $θ$, called the Lee form, such that $dω=θ\wedgeω$. An LCK manifold with potential is an LCK manifold with a positive Kahler potential on its cover, such that the deck group multiplies the Kahler potential by a constant. A Lee class of an LCK manifold is the cohomology class of the Lee form. We determine the set of Lee classes on LCK manifolds admitting an LCK structure with potential, showing that it is an open half-space in $H^1(M,{\mathbb R})$. For Vaisman manifolds, this theorem was proven in 1994 by Tsukada; we give a new self-contained proof of his result. |
| title | Lee classes on LCK manifolds with potential |
| topic | Differential Geometry Algebraic Geometry 53C55, 32G05 |
| url | https://arxiv.org/abs/2112.03363 |